A pipe coupling is known from U.S. Pat. No. 1,932,427, which discloses a flush coupling with a coupling member (pin member 12), the threaded ends of which, developed as spigot elements, have a two-step conical thread. The pipe end which is developed as bell element (box 17) has an outer shoulder and an inner shoulder which lie, in screwed-together condition, in sealing fashion against the corresponding shoulders of the coupling member which is developed in a manner complementary thereto. The disadvantage of this solution is the strong radial inner thickening in the upset head of the pipe which greatly impairs the dynamic behavior of the coupling. As a result, this coupling is not suitable for extremely deep holes. Furthermore, upon installation an elevator nipple must be screwed onto each drill pipe and removed again after the lowering of the pipe. An alternatively possible clamping device on the outside of the flush connection for the holding of the pipe string during the lowering easily leads to damage to the outer surface thereof and thus reduces the life of the drill pipe.